Inverse involute function

Guys,I was trying to find the inverse involute function - inv(alpha)=tan(alpha)-alpha, this funcion will be: alpha=f(inv(alpha)).

I got a approximate one from my friend, which is: (3T)(1/3)-0.4T+0.10697574T(5/3)-0.01648285T(7/3) the fraction in parenthesis is exponent, "T" is inv(alpha). It can get the precise to 5th digital when the angle is under 35 degrees, when alpha is too low, it will also give the wrong result. You can try it. I believe this formula is get from progression of math, but I failed to figure it out myself. I want it to be more precise.

And I made a small function in Excel. It can get the precise to the 10th digital from operate it twice. If you want it, mail to me.

If you can not understand my text, mail to me. Sorry for my poor English.